Abstract

Abstract Let R be a semiprime ring with extended centroid C and with symmetric Martindale quotient ring Q. For a derivation δ of R and an ideal I of R, define Ker I ( δ ) = def . { r ∈ I | δ ( r ) = 0 } and R I ( δ ) = def . δ ( I ) . Let δ , δ ′ be derivations of R such that δ is C-integral and has the associated X-inner derivation ad ( a ) , where a ∈ Q . The main results of this paper (Theorem 3.5) are the following two equivalences: (1) Ker I ( δ ) ⊆ Ker R ( δ ′ ) for an essential ideal I of R if and only if δ ′ = ∑ i ⩾ 1 μ i δ i + ad ( b ) for some μ i ∈ C and some b ∈ C [ a ] . (2) R I ( δ ′ ) ⊆ R R ( δ ) for an essential ideal I of R if and only if δ ′ = ∑ i ⩾ 1 μ i δ i + ad ( b ) for some μ i ∈ C with ∑ i ⩾ 1 ( − 1 ) i δ i ( μ i ) = 0 and some b ∈ C [ a ] .

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